EP1158312A1 - Modelling method for fluids in a fractured environment which is traversed by large fractures - Google Patents

Abstract

Simulating fluid flow in a subterranean reservoir comprises modelling a network of volume elements. Each element includes a fissure and matrix and the fluid flow between them is determined. The reservoir includes a network of fluid conductors having a known geometry but which are not homogeneous at the scale of the volume elements. The overall transmissivity of each conductor is determined from the transmissivity of the conductor within each of the elements it crosses. When the reservoir is in very permeable sedimentary rock, the transmissivity value depends on the size of each element and its contact area with adjacent elements. The transmissivity of each fissure depends on the size of the element, the thickness of the fissure and its intrinsic permeability. The reservoir may be simulated by a transform comprising a network of uniform blocks separated by a network of regular fissures.

Description

The present invention relates to a method for modeling flows fluids in a cracked medium crossed by relatively large fractures, at meaning that we will specify below.

The method can be implemented for example in the field of oil production by field engineers to obtain predictions reliable flow rates in petroleum tanks with discontinuities structural or sedimentary whose flow properties are contrasted by relation to the surrounding medium.

By this modeling, it is a question of simulating flows in a medium porous permeable (reservoir) crossed by a network of cracks and / or layers fines much more conductive than the porous matrix. Cracked tanks constitute an extreme type of heterogeneous reservoirs comprising two media contrasting, a matrix medium containing most of the oil in place and exhibiting low permeability, and a cracked medium representing less than 1% of oil in place and highly conductive. The cracked medium itself can be complex, with different sets of cracks characterized by their density, length, orientation, inclination and respective opening.

In a tank in production, the pressure and flow conditions imposed on the producing and injecting wells induce fluid flows in place in the tank (oil, gas and water). The simulation of these flows consists to determine the evolution of pressures and saturations in the reservoir during the time. Due to the very heterogeneous nature of fractured reservoirs, fluids are move relatively quickly through the network of fractures and much more slowly into the matrix. The simulation of flows in the context of a fractured reservoir therefore requires a very good representation of heterogeneities major fractures. The accuracy of this representation depends on the type of simulation implemented and the size of the fractures to be modeled. So, for a very precise simulation of a well test, we will have to represent the exact geometry of the fracture network. Conversely, for taking into account a dense network of small fractures on the scale of a field, we will seek a equivalent simplified representation.

The present invention makes it possible to simulate precisely the flows at the scale of a field in the presence of many large fractures.

STATE OF THE ART

Currently in the petroleum industry in particular, models are used. double porosity (double medium) to simulate flows in media fractured but they are applied not to the actual geological medium in all of its complexity but with a homogenized representation, according to the reservoir model said of double medium described for example by Warren and Root, in "The Behavior of Naturally Fractured Reservoirs ”, SPE Journal, 1963. Any elementary volume of the cracked tank is considered to consist of two homogeneous media equivalent to the mesh scale of the simulator: a crack medium and a medium matrix. The flow of fluids on the tank scale is mainly carried out at through the crack medium and fluid exchanges occur locally between the cracks and matrix blocks. This representation which does not reflect the complexity of the network of fractures in a reservoir, is nevertheless effective at the rung of a tank mesh typically having the dimensions 100m x 100m. Dual medium modeling allows to reproduce the behavior at the flow of the two media and their interactions without having to resort to an explicit modeling of the two environments.

By patent FR-A-2,757,947 (US 6,023,656) of the applicant, we know a technique for determining the equivalent crack permeability of a network of cracks in an underground multi-layer medium from a known representation of this network. It allows to systematically link models of characterization of a cracked reservoir with dual porosity simulators in order to more realistic modeling of a cracked underground geological structure.

By patent FR-A-2,757,957 of the applicant, we know a technique for obtain a simplified modeling of a porous heterogeneous geological medium (such that a reservoir crossed by an irregular network of cracks for example) under the form of a transposed medium or equivalent so that the transposed medium is equivalent to the medium of origin, relative to a determined type of function of physical transfer (known for the transposed environment).

By patent application FR 98 / 15.727 of the applicant, we also know a method for modeling the flow of fluids in a porous multi-layer medium cracked taking into account the real geometry of the network of cracks and exchanges premises between the porous matrix and the cracks in each node of the network. We discretizes the cracked medium by a mesh, one centers the meshs of crack on nodes at the different intersections of the cracks, each node being associated with a volume of matrix, and one determines the fluxes between each mesh of crack and the volume of associated matrix in a semi-permanent flow regime.

There are cases where the above techniques are difficult to implement work, where one is in the presence of a medium crossed by large fractures or sub-seismic faults whose hydraulic behavior cannot be homogenized at the scale of the mesh. Explicit modeling of these objects is then necessary to a priori but their number too high would make such an approach prohibitive on the scale of a field (too many meshes and numerical constraints). The same problem arises for tanks containing thin and very permeable benches whose behavior is similar to that of large horizontal fractures.

The method according to the invention

The modeling method according to the invention makes it possible to simulate fluid flows in a cracked porous geological medium of known structure that we discretize by a mesh and that we model by considering that each elementary volume of the cracked geological medium consists of a crack medium and of a matrix medium equivalent to the scale of each mesh between which one determines the exchange of fluids, this geological medium being crossed by a network of conductive objects of fluids of defined geometry but not homogenizable with the scale of each mesh of the model (large fractures, sub-seismic faults for example, very permeable sedimentary layers, etc.). The method involves a determination of the exchanges occurring between the matrix medium and the middle crack, and a modelization of the transmissivities of the various meshes traversed by each conductive object, so that the resulting transmissivity corresponds to direct transmissivity along this conductive object.

In the case where the conductive objects are very sedimentary layers permeable, we attribute to the transmissivity between meshes crossed by each layer very permeable a value depending on the dimensions of the meshes and the area of common contact between layers at the junction of adjacent meshes.

In the case where the conductive objects are fractures, we attribute to the transmissivity between meshes crossed by each fracture, a transmissivity depending on the dimensions of the meshes and the common area of the fracture at the junction of adjacent meshes.

• la détermination, pour chaque pixel, de la distance minimale qui le sépare de la fissure la plus proche ;
• la formation d'une distribution du nombre de pixels par rapport à la distance minimale au milieu fissuré et la détermination, à partir de cette distribution, de la fonction de récupération (R) dudit ensemble de blocs ; et
• la détermination de dimensions des blocs réguliers équivalents du milieu transposé à partir de la récupération (R) et de la récupération du bloc équivalent.

In all the meshes crossed by geometrically defined conductive objects (matrix blocks of irregular shapes and sizes), a transposed medium is determined comprising a set of blocks regularly arranged and separated by a regular mesh of cracks giving substantially the same recovery function of fluid during a capillary imbibition process than the real medium. The vertical dimension of the blocks of the transposed medium is calculated from the positions of the very permeable sedimentary layers in the mesh and the horizontal dimensions of the blocks of this transposed medium are obtained from a two-dimensional (2D) image of the geological medium. as a series of pixels, by:

• the determination, for each pixel, of the minimum distance which separates it from the nearest crack;
• forming a distribution of the number of pixels with respect to the minimum distance from the cracked medium and determining, from this distribution, the recovery function (R) of said set of blocks; and
• determining the dimensions of the equivalent regular blocks of the transposed medium from the recovery (R) and the recovery of the equivalent block.

Presentation of the figures

• la Fig.1 montre schématiquement deux mailles adjacentes d'une même couche d'un maillage de réservoir où la présence de niveaux sédimentaires fins et très perméables induit une transmissivité horizontale dans le milieu « fissure » ;
• la Fig.2 montre un maillage de réservoir traversé par un réseau de fissures ;
• la Fig.3 montre des mailles O, A, B, C d'un réservoir traversée par une fissure que l'on modélise par des transmissivités fissure-fissure ;
• la Fig.4 illustre le mode de calcul de la transmissivité entre deux mailles A et B traversées par une fissure ;
• la Fig.5 montre par comparaison le trajet en marches d'escalier au travers de mailles traversées par une fracture oblique, que l'on prend en compte pour les besoins de la simulation, et dont la transmissivité, selon le mode de calcul choisi, est cependant équivalente à la transmissivité réelle directement le long de la fracture ; et
• la Fig.6 illustre le mode de calcul de la taille d'un bloc équivalent en fonction du nombre de niveaux sédimentaires très perméable traversant une maille ;
• la Fig.7 présente un exemple de pixels voisins servant au calcul de la valeur attribuée à un pixel ; et
• la Fig. 8 présente une variation possible d'une zone envahie normée en fonction de la distance aux fissures.

Other characteristics and advantages of the method according to the invention will appear on reading the following description of a non-limiting example of embodiment, with reference to the appended drawings where:

• Fig.1 schematically shows two adjacent meshes of the same layer of a reservoir mesh where the presence of fine and very permeable sedimentary levels induces horizontal transmissivity in the "crack"medium;
• Fig.2 shows a reservoir mesh crossed by a network of cracks;
• Fig.3 shows meshes O, A, B, C of a reservoir crossed by a crack which is modeled by crack-crack transmissivities;
• Fig.4 illustrates the method of calculating the transmissivity between two meshes A and B crossed by a crack;
• Fig. 5 shows by comparison the path in staircase steps through meshes crossed by an oblique fracture, which is taken into account for the needs of the simulation, and whose transmissivity, according to the chosen calculation method, is however equivalent to the real transmissivity directly along the fracture; and
• Fig.6 illustrates the method of calculating the size of an equivalent block as a function of the number of very permeable sedimentary levels crossing a mesh;
• Fig.7 shows an example of neighboring pixels used to calculate the value assigned to a pixel; and
• Fig. 8 shows a possible variation of a normalized invaded area as a function of the distance to the cracks.

DETAILED DESCRIPTION

We will consider below the example of a porous reservoir crossed by a network of fractures F (Fig. 2) which are assumed to be vertical to simplify and by fine sediment levels (sub-horizontal) L (Fig. 1) whose properties petrophysics (notably permeability) are contrasted with respect to the host matrix. This tank is modeled in the form of two "superimposed" grids (double medium model), one of them, qualified as “matrix”, representing the matrix, the other, qualified as a “crack”, representing all of the discontinuities considered (fractures and fine permeable levels). The flows are calculated within the matrix grid and the crack grid respectively; of more, terms of exchange connect the unknown factors of each pair of matrix meshes and cracking of the model by means of suitable formulations. The method that is going to be described makes it possible to calculate the transmissivities between “crack” meshes and the exchanges "Matrix-crack". The exchanges between the matrix meshes are calculated so standard well known to those skilled in the art.

I-Transmissivities between “crack” meshes I-1 Transmissivities associated with fine sedimentary levels and permeable

Fine and permeable sedimentary levels are included in the environment "Crack" of the double middle model. In a mesh crossed by such levels sedimentary, the petrophysical properties of these levels (porosity, permeability, water saturation) are affected in the middle crack of the mesh and the properties of rest of the rock contained in the mesh are affected in the middle matrix of the same mesh.

The presence of fine and very permeable sedimentary levels in two adjacent meshes induces horizontal transmissivity in the “crack” medium between the two meshes (“crack-crack” transmissivity). The diagram in Fig. 1 shows two adjacent meshes (in the same layer of the reservoir mesh) containing such levels. We note that there is no vertical transmissivity Induced "crack-crack" since the levels are horizontal.

Ks est la perméabilité des niveaux sédimentaires très perméables,

• Y est la taille des mailles en Y,
• X est la taille en X, et

Es_i,i+1 est l'épaisseur des contacts entre niveaux sédimentaires fins et très perméables des deux mailles adjacentes i et i+1.

In this example, the horizontal “crack-crack” transmissivity between the meshes i and i + 1 is calculated as follows: Ts i, i + 1 = Ks · Es i, i + 1 ΔY ΔX , or

Ks is the permeability of very permeable sedimentary levels,

• Y is the size of the meshes in Y,
• X is the size in X, and

Es _{i, i + 1} is the thickness of the contacts between fine and very permeable sedimentary levels of the two adjacent meshes i and i + 1.

This thickness is zero if the levels of the two meshes are not connected. Conversely, if they are fully connected, it can be equal to the most small of the cumulative thicknesses of levels in the two meshes.

I-2 Transmissivities associated with fractures

The network of vertical fractures is also taken into account in the environment "Crack" of the double middle approach. In each layer of the reservoir model (Fig. 2), this network can be represented by a series of crack segments which cross the reservoir mesh, as in the following diagram:

• la perméabilité fissure Kf,
• l'épaisseur de la fissure Ef, et
• la longueur de fissure Lf.

The data relating to these crack segments are:

• crack permeability Kf,
• the thickness of the crack Ef, and
• the crack length Lf.

The crack porosity • f, calculated by the following formula: Φf = Lf · Ef ΔX · ΔY

The communication between the cells of the reservoir through the network of fractures is modeled by crack-crack transmissivities. On the example of Fig.3, crack-crack transmissivities are calculated between the meshes crossed by the fracture, i.e. for the couples O and A, A and B and B and C: T _FOA , T _FAB and T _FBC .

This example shows that the actual path of a flow through a fracture can to be far from the way imposed by the modeling which passes by the centers of the meshes O, A, B and C. The solution which would consist in replacing the fracture segment by a broken line passing through the mesh centers would lead to a bad simulation flows through these meshes.

Kf est la perméabilité intrinsèque de la fracture,

Ef est l'épaisseur de la fracture,

•Z est l'épaisseur de la couche,

L_ab est la longueur du segment ab,

a est le milieu du segment de fracture traversant la maille A, et

b est le milieu du segment de fracture traversant la maille B

Also, one determines the horizontal “crack-crack” transmissivity between two meshes crossed by the same fracture (cf. Fig.4), as follows: T FAB = Kf · Ef · ΔZ L ab or:

Kf is the intrinsic permeability of the fracture,

Ef is the thickness of the fracture,

• Z is the thickness of the layer,

L _ab is the length of segment ab,

a is the midpoint of the fracture segment crossing mesh A, and

b is the middle of the fracture segment crossing mesh B

It is noted that the smaller the length L _ab , the greater the “crack-crack” transmissivity between the meshes A and B. Thus, we digitally correct the grid effect which imposes a staircase path on the flow.

Thus the flow between two distant meshes connected by a fracture can be correctly simulated despite the staircase path (Fig.5) imposed by the grid. Indeed, on the example of FIG. 5, we can verify that:
Σ 1 / T _fi = 1 / T _fMN where the T _fi are the “crack-crack” transmissivities between Cartesian reservoir cells along the staircase path connecting M and N, and T _fMN is the real transmissivity of the fracture between M and N.

Kf est la perméabilité intrinsèque de la fracture,

Ef est l'épaisseur de la fracture,

•Z est l'épaisseur de la couche,

L_f est la longueur du segment de fracture dans la maille considérée.

Concerning the vertical “crack-crack” permeability induced by the presence of a fracture crossing several layers of the reservoir, we have: T fV = Kf · Ef · Lf ΔZ or :

Kf is the intrinsic permeability of the fracture,

Ef is the thickness of the fracture,

• Z is the thickness of the layer,

L _f is the length of the fracture segment in the considered mesh.

Finally, when several fractures cross a mesh, the transmissivities calculated for these individual fractures are added.

I-3 Resulting transmissivity

The transmissivities relating to the highly permeable sedimentary levels (T _s ) and the transmissivities relating to the fracture (T _f ) are calculated separately according to the methods described above. The “crack-crack” transmissivities of the final double medium model are calculated simply by adding T _s and T _f .

Likewise, the final porosity of the “crack” medium in each mesh is the sum of the porosities due to the very permeable sedimentary levels on the one hand and fractures on the other hand.

II- Dimensions of equivalent matrix block II-1 Horizontal dimensions

The horizontal dimensions of the equivalent block (a, b) are controlled by the vertical fractures in the tank. Indeed, the “matrix-crack” exchanges »Due to vertical fractures are only horizontal.

In each mesh crossed by at least one fracture, these dimensions horizontal are calculated by the method described in patent FR 2,757,957 cited above. For meshes in which there is no fracture, the dimensions equivalent block horizontals are infinite. In practice, we assign them a value very large (eg 10 km). According to this method we determine the dimensions of equivalent block by identifying the behaviors of the real fractured medium and the equivalent medium for a two-phase water-oil imbibition mechanism. That consists in matching the oil recovery function R (t) (of the real fractured medium) obtained by an image processing method (described more bottom) and of the recovery function Req (t) of the equivalent medium whose expression analytical is known and depends on the dimensions of the equivalent block.

II-1-a) Geometric formulation

The cracks being defined by the coordinates of their limit points on a two-dimensional XY cut of a layer, the process of imbibition by which of water is present in the cracks and oil is present in the matrix blocks must be determined. We suppose that the invasion of the matrix by water is of the type piston. We admit that the function x = f (t) which links the advance of the water front to time is the same for all matrix blocks, whatever their shape, and for all elementary blocks. Consequently, the matching of the functions R (t) and Req (t) is equivalent to matching the functions R (x) and Req (x). These functions physically define standardized areas overgrown with water based on the advance of the imbibition front in the cracked medium.

où a et b sont les dimensions du bloc rectangulaire ou carré équivalent (a et b > 0) :In 2D, the analytical expression of Req (x) is as follows:

where a and b are the dimensions of the equivalent rectangular or square block (a and b> 0):

The function R (x) has no analytical expression. It is calculated from of a discretization of the XY cut of the studied layer according to the defined algorithm below.

II-1-b) Algorithm for calculating the function R (x)

The XY section of the studied layer is considered as an image whose each pixel represents a surface element. These pixels are evenly spaced of a step dX in the direction X and dY in the direction Y (Fig. 7). The algorithm put implemented aims to determine, for each pixel of this image, the minimum distance which separates it from the nearest crack.

The image is translated by an array of two-dimensional real numbers: Pict [0: nx + 1.0: ny + 1] where nx and ny are the pixel numbers of the image in the directions X and Y. In practice, the total number of pixels (nx.ny) is for example of the order of a million. The values of the elements of the Pict array are the distances wanted.

Initialization: All the pixels through which a crack passes are at a zero distance from the nearest crack. For these pixels, the table Pict is therefore initialized to the value 0. This is done by an algorithm known per se (the Bresline algorithm for example) to which one gives the coordinates of the pixels corresponding to the two ends of a crack considered as a line segment and which initializes (to 0 in this case) the nearest pixels. The other elements of Pict are initialized to a value greater than the greatest distance existing between two pixels of the image. This value is for example nx.dX + ny.dY.

Computation: For a given pixel, the computation of the distance sought with the nearest crack is done starting from the values of distance already calculated for the neighboring pixels. It is assigned a value which, if it turns out to be less than the value which was initially assigned to it, is the minimum of the values of the neighboring pixels to which the distance of these pixels is added to that considered.

This calculation is carried out in two successive phases. During the pass descending, we browse the image line by line, from top to bottom and from left to right (from Pict [1,1] to Pict [nx, ny]). The pixels taken into account are different depending on whether one is in a downward pass or an upward pass. As the Fig. 7, the pixels in black and grayed out are those that are taken into account respectively during the descending passes and the ascending passes, for the pixel Px.

The oblique deviation dxy being defined as: dxy = dx 2 + dy 2 , the algorithm is written:

Histogram: From the Pict table thus calculated, we can draw a histogram by classifying the non-zero values (those assigned to the pixels outside the cracks) in ascending order.

The cumulative of this histogram gives, for any distance delimiting two histogram intervals, the number of non-zero pixels whose value is less than this distance. In the application described to a cracked porous medium where this distance corresponds to the advance of the water front, the cumulative of the histogram therefore indicates the area invaded by water. The curve R (x) is obtained by dividing this accumulated by the total number of non-zero pixels (to normalize it). The number of intervals used on the abscissa for the histogram corresponds to the number of points of discretization of the curve R (x). We choose it equal to 500 for example.

II-1-c) Finding the dimensions of the equivalent block

où N est le nombre de points de discrétisation de R(x) et (x_i) sont les abscisses de ces points de discrétisation.At this stage, we know the function R (x) and we look for the parameters ( at , b ) (dimensions of the equivalent block which minimize the functional):

where N is the number of discretization points of R (x) and (x _i ) are the abscissas of these discretization points.

Discretization according to the ordinate of R (x):

To give as much weight to all the volumes of oil recovered during imbibition, the curve R (x) is re-discretized according to a constant step on the ordinate axis (Fig. 8). The sequence (x _i ) used by the functional is deduced from this discretization.

Minimization of the functional:

As a and b play symmetrical roles in the expression Req (a, b, x), we actually use the functional:

(u,v) = 0. Ceci est fait par un algorithme de Newton.Minimizing this functional means finding the couple ( u , v ) for which

( u , v ) = 0. This is done by a Newton algorithm.

1) v > 0 signifie qu'une des valeurs du couple (a,b) est négative, ce qui n'a aucun sens physique. On pose alors v=0 dans l'expression de R

q(u,v,x), ce qui implique que les fissures sont parallèles. L'opération est recommencée et le couple (a,b) est calculé comme suit :

2) Le cas u ² + 4v < 0 est dépourvu de tout sens physique également puisqu'il signifie que (a,b) ne sont pas réels. On pose alors u² + 4v = 0, ce qui impose que le bloc élémentaire recherché a la forme d'un carré (a=b). Après minimisation, le couple (a,b) est calculé comme suit :

3) Pour les autres valeurs du couple (u,v), on a :

Then the couple ( at , b ) sought is deduced from ( u , v ). Three cases can arise:

1) v > 0 means that one of the values of the couple ( at , b ) is negative, which has no physical meaning. We then set v = 0 in the expression of R

q (u, v, x), which implies that the cracks are parallel. The operation is restarted and the couple ( at , b ) is calculated as follows:

2) The case u ² + 4 v <0 is also devoid of any physical meaning since it means that ( at , b ) are not real. We then set u ² + 4v = 0, which requires that the elementary block sought has the shape of a square (a = b). After minimization, the couple ( at , b ) is calculated as follows:

3) For the other values of the torque ( u , v ), we have :

II-2 Vertical dimension

The vertical dimension (c) of the equivalent block is controlled by the levels horizontal sediment, fine and very permeable (Fig. 6). The “matrix-crack” exchanges »Resulting from these levels are indeed only vertical.

In each mesh crossed by at least one very permeable level, the vertical dimension of the equivalent block is calculated by the following formula: c = ΔZ Ns where • Z is the thickness of the mesh and Ns is the number of distinct very permeable sedimentary levels in the mesh. For example, on the following diagram, Ns is worth 2:

For meshes in which there are no very permeable fine levels, the equivalent block height is infinite. In practice, they are also assigned a very great value (eg 10 km).

Claims (4)

Modeling method making it possible to simulate fluid flows in a cracked porous geological medium of known structure which is discretized by a mesh and which is modeled by considering that each elementary volume of the cracked geological medium consists of a crack medium and of a matrix medium equivalent to the scale of each mesh between which the exchange of fluids is determined, this geological medium being crossed by a network of conductive objects of fluids of defined geometry but not homogenizable on the scale of each mesh of the model, comprising a determination of the exchanges occurring between the matrix medium and the crack medium, characterized in that one models the transmissivities of the various meshes crossed by each conducting object, so that the resulting transmissivity corresponds to direct transmissivity along of this conductive object. Method according to claim 1, characterized in that , in the case where the conductive objects are very permeable sedimentary layers, the transmissivity between adjacent meshes crossed by each very permeable layer is assigned a value depending on the dimensions of the meshes, of the common contact area between layers at the junction of adjacent meshes and the permeability of highly permeable layers. Method according to claim 1, characterized in that , in the case where the conductive objects are fractures, the transmissivity between adjacent meshes crossed by each fracture is attributed a transmissivity depending on the dimensions of the meshes, on the thickness of the fracture and its intrinsic permeability. Method according to one of the preceding claims, characterized in that in the meshes crossed by geometrically defined conductive objects, a transposed medium is determined comprising a set of blocks regularly arranged and separated by a regular mesh of cracks, said transposed medium giving substantially the same recovery (Req) of fluid during a capillary imbibition process as the real medium, the vertical dimension of the blocks of the transposed medium being determined from the positions of the very permeable sedimentary layers in the mesh and the horizontal dimensions of the blocks of this transposed medium being obtained, from a two-dimensional (2D) image of the geological medium in the form of a series of pixels, by: determining, for each pixel, the minimum distance separating it from the nearest crack; forming a distribution of the number of pixels with respect to the minimum distance from the cracked medium and determining, from this distribution, the recovery function (R) of said set of blocks; and determining the dimensions (a, b) of the equivalent regular blocks of the medium transposed from the recovery (R) and the recovery (Req) of the equivalent block.

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